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If z(1), z(2) are complex numbers such t...

If `z_(1)`, `z_(2)` are complex numbers such that `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|` and `arg(z_(1)-z_(2))=pi//3` , then `Im(z_(1)+z_(2))=`

A

`2//sqrt(3)`

B

`4//sqrt(3)`

C

`2//sqrt(3)`

D

`sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` Let `z_(1)=x_(1)+iy_(1)` and `z_(2)=x_(2)+iy_(2)`
Given `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|`
`:.y_(1)^(2)-4x_(1)+4=0` and `y_(2)^(2)-4x_(2)+4=0`
So that `(y_(1)-y_(2))/(x_(1)-x_(2))=(4)/(y_(1)+y_(2))` ……………`(i)`
Given `arg(z_(1)-z_(2)=pi//3`
`implies (y_(1)-y_(2))/(x_(1)-x_(2))=sqrt(3)` .........`(ii)`
From `(i)` and `(ii) implies y_(1)+y_(2)=(4)/(sqrt(3))`
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